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How Much Life Insurance Do You Need?
Chris Robinson[1] and Victoria Zaremba[2]
March 29, 2014
Abstract
We present formal models of the different methods of estimating a person’s required life insurance coverage. The models make evident the critical issues surrounding the estimates: the family goals, cash flow estimation, the discount rate, inflation, taxes and the time horizons assumed. We investigated 23 life insurance companies to see what models they and/or their agents use to estimate life insurance needs of potential clients. In general, where we could determine a model, it was closest to the expense model. The models and the explanations are rather vague about the details and application of them could lead to widely-varying results. We found similar vagueness in the models that appear in life insurance textbooks. This great lack of attention to how life insurance needs are calculated suggests that personal financial planning education and the insurance industry should be working to improve both teaching and practice on this topic.
Simple Models for Estimating Life Insurance Needs
There are two common methods of estimating the amount of life insurance a person needs, assuming that there will be surviving family members who are partially or wholly dependent upon the earnings of the insured person. The income method calculates the present value of all the future labour income of the insured, sometimes with a rule of thumb adjustment to allow for items like income tax rate differences and saved expenses. The expense method calculates the present value of all the future expenses of the dependents that the labour income was intended to support. In between the income and expense methods is the human live value or what we call the net human capital method, which we will argue is likely to serve family needs better than either the income or the expense method. A different approach values the insurance at the amount needed to produce a specific bequest.
Let us establish some notation so that we can express these concepts more formally.
PNIAdditional insurance required using the income method
PNXAdditional insurance required using the expense method
PNHAdditional insurance required using the net human capital method
PCPAdditional insurance required using the capital retention method
PEExisting insurance coverage
EtAfter-tax real dollar labour income of the insured, year t
CtReal dollar consumption of the family with the insured person alive, year t
Real dollar consumption of the family after the insured’s death, year t
kafter-tax real discount rate
knomafter-tax nominal discount rate
Rdate of retirement
Ddate of death last surviving dependent, or end date of dependence on the insured
W0Initial wealth available to fund future consumption of the dependents
WnFuture value of a family’s financial goal
BDBequest to the next generation
Then, under the income method,
(1)
Under the expense method,
(2)
The family’s consumption after the death of the insured person needs some further description. We are defining family as a surviving spouse for the rest of his or her life, and the children or others only for as long as they would normally remain dependent on the spouse. Once the children are expected to start earning their own income and paying their own expenses, they are no longer part of the family, but have started their life as family units themselves. The expenses that need to be insured no longer include their expenses. The actual age of a child when independence happens can vary, and it might be for life in the case of a disabled child.
The two expressions are quite different, but what does the difference mean? If you insure your after-tax income, including pension contributions, then you must have at least covered all the expected expenses, because otherwise you would also not have been able to cover them if you didn’t die before retirement. We expect PNI ≥ PNX.
We can say more than that. In a world of certainty, we would be able to calculatePNX precisely, and if the insured died, the insurance money would run out exactly on the date of death of the second spouse. The difference between the income and expense methods of insurance is the bequest to the next generation, or more precisely, the additional bequest to be added by the new life insurance. We can express this as:
(3)
In this form, as the difference between the two methods of calculating life insurance needed today, the equation gives the present value of the bequest. At the date of death, the bequest will be:
(4)
In a more general sense, we can express the wealth, or perhaps the goal, of a family, at any time n, as:
(5)
If the specific goal is the bequest to the next generation, then in the absence of any insurance and with neither parent dying prior to retirement, we get:
(6)
Equations (4) and (6) do not yield the same value for BDin general, though they could do so in specific circumstances. They appear very similar, but we have glossed over a number of important issues that affect the value that life insurance will have for different families in different circumstances. We have also treated the problem only in a certain world, but life insurance is all about risk management, and we cannot ignore the uncertainty in these estimates. One of the biggest effects is income tax. A common practice with the income method is to reduce the basic calculation to 70 or 80% of the value, as a rule of thumb adjustment for tax and risk effects.
The insurance literature focuses much more on concerns related to actuarial measurements than it does to the micro-level of how much life insurance a specific family needs. When the economics and finance literature turns to this question, the expense method is preferred, though the analysis we have presented does not appear.
Bernheim et al. (1999, pg. 6), in a paper measuring the extent of underinsurance, say:
…life insurance is defined to be adequate if the survivor’s highest sustainable standard of living after the death of a spouse is equal to or greater than the couple’s highest sustainable standard of living if both survive. We equate standard of living with consumption, adjusted for household demographic composition. In other words, insurance adequacy, in the sense that we use the term here, is associated with demographically-adjusted consumption smoothing across the states of nature associated with the survival or death of the covered spouse.
Naylor et al. (2013) explicitly adopt an expense model in their work on underinsurance. They interpret the expense model and delve more into the details of what expenses are after death of a spouse than do other researchers. In this regard, their work is similar to the model we propose in the next section.
We argue that the choice of method should match the family goals and a more precise consideration of what the human capital is worth and how much the second generation can expect as the average bequest.
Net Human Capital Model
Let us step back and consider the purpose of insurance. What is the family objective in buying life insurance? One objective is often said to be maintaining the family in the same life style as if the insured did not die prematurely and continued to contribute to the family resources. Let us reason by analogy and think of house insurance. The usual practice is to insure the entire value of the family home and either add a current value rider or revise the insurance periodically to reflect changes in the replacement value of the home. The purpose of the insurance is to replace the entire home with the same quality home. This home insurance does not have the same objective as the one we stated for life insurance. Recall from our previous discussion that the family consists of the dependents only as long as they are dependent. Later in the life cycle, this will be only the parents most often, because the children will have moved on to form their own families. If the objective in insuring the house is only to maintain the standard of living required for the dependent children and surviving spouse, it could be met more cheaply by insuring for an amount equal to the present value of the rental cost of an equivalent house until the death of the surviving spouse. But we do not think of house insurance in that way, we think of it as replacing the entire value of what the family already owns, the house and land freehold, with the right to sell it for market value upon death of the spouse, and leave the residual to the next generation. Or alternatively, to sell the house when the surviving spouse is older and needs the asset value to generate income.
If we apply the logic of house insurance to life insurance, we want to insure the entire value of the asset, which is the human capital. This seems to resolve the question between the income and expense methods in favour of the income method, because it is a way of measuring human capital. However, we recall also one maxim favoured by a colleague, which is that you should never insure yourself so well that the beneficiary is better off with you dead than alive! There are two reasons why the pure income method will also fail to capture the value to the family of the human capital of the insured.
First, recall that we don’t value a business solely based on its revenue (which equals labour income in the human capital sense), but rather on its net income or cash flow. Human capital has to be maintained with food, clothes, etc., and when the insured dies, these expenses also disappear. The net contribution to the family is much less than the labour income.
Second, human beings are not valuable solely for their cash income, but also for the value they contribute to the family. A common mistake in family insurance plans is to fail to insure the spouse who stays home with the children, yet if he or she dies, the survivor will have significant additional expenses. Spouses working outside the home also contribute labour to the household that would otherwise have to be purchased.
In words, the net human capital method of estimating insurance needs measures the net contribution to the family of the person’s human capital by valuing the paid labour income, the unpaid labour income and the savings in maintenance of the human capital. Huebner (1927) seems to be the first to discuss insurance needs in this fashion, but he does not provide a detailed model of how to calculate what he calls human life values. He provides an example that calculates the present value of the expected future earnings of a person, minus the cost needed to care for the insured person if he had lived (Huebner, 1927, pp. 37-38). He discusses some of the issues around the estimation, but in fairly general terms. He does not deal with the insured person’s contributions to household work, probably because he implicitly assumes a household with a working male and homemaking female. Our definition of net human capital is more inclusive.
We can express the net human capital value as:
(7)
Note that the summation for the labour income runs until the retirement of the insured, but the summation of the net saving or net extra expenses required runs until the death of the spouse. The sign of each term in the summation will not necessarily be the same. If the insured is a homemaker spouse, the death could increase expenses while the children are dependent and then decrease them after the departure of the children.
We expect PNH ≥ PNX also, but the relationship to PNI depends on the circumstances. A homemaker who earns no outside labour income will seem to require no life insurance, but a proper calculation of his or her contribution to the family will show a positive value. A family in which both spouses work outside the home and share the household duties or hire others to do them will be one in which PNI≥ PNH ≥ PNX.
Naylor et al. (2013) say they use the expense method, but the detailed description of their work seems closer to what we are calling the net human capital method.
Specific Bequest Models
A bequest to a spouse is simply the provision for expenses, or the capitalisation of net human capital and should not be confused with the models in this section. The specific bequest models are providing a bequest to the next generation. We have already seen that the simple net income model provides an expected bequest to the next generation that is equal what the next generation would expect to receive if the insured person did not die prematurely.
There are two types of specific bequest calculations. In both cases, the implicit family goal is dynastic. The family wants to maintain specific assets, or perhaps just a fortune, forever. One such model is the capital retention approach. The insurance allows the dependents to live upon the income from the family capital without liquidating any of it, thus passing an estate to the next generation with virtual certainty. The authors who describe this method do not distinguish between maintaining the real value of the estate and the nominal value.
If we take the goal as maintaining the real value, then,
PCP = C1/k - PE(8)
Comparing this to the earlier equations, we see that the initial wealth is not counted at all in the determination of needed insurance, because that is precisely what the family wants to maintain. If instead the family believed that the insured person would be able to increase the family fortune materially if he or she lived to the end of working life, then there would be an additional term that is the capitalised value of expected future excess income. If the goal is to maintain the nominal value, then we substitute knom for k.
The other sort of specific bequest allows the family to keep a specific asset that the heirs would otherwise have to sell in order to pay income taxes or death duties. In Canada, assets generally can pass from one spouse to the other at death without immediate tax consequences, but the capital gains tax will be exacted when the second spouse dies. If the asset is something that has special value outside its financial worth, like a family cottage or a family business in which other family members still work, then the desired value of the insurance is the estimate of the future income tax that will be payable.
Issues in the Estimation Process
We identify six issues that may materially affect the estimate of life insurance needed:
- The family’s goal in buying the insurance;
- The estimation of future earnings and living expenses;
- Time horizons that are assumed, particularly date of death of the beneficiary spouse and retirement age of the insured;
- Riskless or risky discount rate;
- Inflation; and,
- Income tax.
These issues are inter-related, but we will discuss them separately, while showing the connections.
Family Goal
It follows from the characteristics of the three models that the method used to estimate life insurance needs depends upon the family goals.
Does the family wish to leave a bequest to the next generation, assuming it even has the means to do so? If the answer is no, then the insured parents should use the expense method, since this will not produce as much of a bequest to the next generation, if at all.
If the family wishes to leave a bequest, it could use thenet human capital method. This method will on average reproduce the bequest that the next generation could expect if neither parent died early. The income method will produce an excessive bequest on average, although a rule of thumb adjustment might correct it by accident.
Another method is possible, though. If the goal is to leave a bequest, the insured person could set a specific bequest goal, and it could be less than the amount that the net human capital method would produce. The mechanical calculation is to use the expense method to determine the needed insurance, and then add as much insurance as is desired for the bequest. This amount would be in nominal dollars at the date the insurance is purchased and so the real value of the bequest would be lower. If the bequest is intended to be specifically dynastic, then the insurance is estimated as the amount needed to maintain whatever it is that the family wants to be able to keep in perpetuity.